Imagining Solids

Transcript

Imagining SolidsWe start with a quote from Nikola Tesla My method is different. I do not rush into actual work. When I get an idea I start at once building it up in my imagination. I change the construction, make improvements, and operate the device entirely in my mind. Nikolas Tesla (1856-1943), the great Serbian-American engineer and inventor, who made fundamental contributions to electrical engineering and other fields. Why put an electrical engineer in a math book? Because math isn't just about writing formulas on paper—it is about training your brain to be a simulator. Tesla was famous for inventing complex motors entirely in his head before ever touching a tool. Before diving into formulas for surface area or volume, this section is training the brain to translate 2D images into 3D mental models, and vice-versa. It is exactly the kind of thinking engineers, architects, and designers use every day! Now, let’s answer some questions. For each question, feel free to talk to your partner, gesture, draw it in the air — but do not actually draw on paper! Question 1: Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend’s name. Let's picture the name DAVNEET. If you close your eyes and truly visualize those letters floating in front of you, reading them backward by sight (not sound) gives you: T - E - E - N - V - A - D. Question 2: Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square? If you have a square and cut off the four corners from midpoint to midpoint, the shape left over is a smaller square (it will look like a diamond, rotated 45 degrees inside the original square). If you take those four triangular corners you cut off and put their straight edges (the right angles) together in the center, they will form another square of the exact same size as the one left over! Question 3: Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get? If you mark an equilateral triangle into thirds and cut off the corners, you are removing three smaller equilateral triangles. The shape left over has 6 sides of equal length. It is a regular hexagon. Question 4: Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left? If you do the same to a square (cutting the corners off at the one-third marks), you remove four triangles. The remaining shape has 8 sides, making it an octagon (Note: It's not a regular octagon, because the slanted sides where you made the cuts will be slightly longer than the straight sides left over).